High order homogenized Stokes models capture all three regimes

Florian Feppon Department of Mathematics - ETH Wenjia Jing Mathematical Sciences Center [Tsinghua]

Analysis of PDEs mathscidoc:2206.03018

SIAM Journal on Mathematical Analysis
This article is a sequel to our previous work [13] concerned with the derivation of high-order homogenized models for the Stokes equation in a periodic porous medium. We provide an improved asymptotic analysis of the coefficients of the higher order models in the low-volume fraction regime whereby the periodic obstacles are rescaled by a factor η which converges to zero. By introducing a new family of order k corrector tensors with a controlled growth as η→0 uniform in k∈\N, we are able to show that both the infinite order and the finite order models converge in a coefficient-wise sense to the three classical asymptotic regimes. Namely, we retrieve the Darcy model, the Brinkman equation or the Stokes equation in the homogeneous cubic domain depending on whether η is respectively larger, proportional to, or smaller than the critical size η_{crit}∼ϵ2/(d−2). For completeness, the paper first establishes the analogous results for the perforated Poisson equation, considered as a simplified scalar version of the Stokes system.
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@inproceedings{florianhigh,
  title={High order homogenized Stokes models capture all three regimes},
  author={Florian Feppon, and Wenjia Jing},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220626171537180632474},
  booktitle={SIAM Journal on Mathematical Analysis},
}
Florian Feppon, and Wenjia Jing. High order homogenized Stokes models capture all three regimes. In SIAM Journal on Mathematical Analysis. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220626171537180632474.
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